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In the 1960s Mandelbrot developed the use of fractals to describe how the shape of many aspects of the natural world departs from the Euclidean. In particular he proposed two kinds of fractal model to capture the way in which natural data is often persistent in time his Joseph effect, common in hydrology and exemplified by fractional Brownian motion and or prone to heavy tailed jumps the Noah effect, typical of economic index time series, for which he gave L\'evy flights as an exemplar. Both effects are now well demonstrated in proxies both for the Earth's auroral electric currents and for the turbulent solar wind which is their ultimate energy source. Modelling, however, has usually emphasised one of the Noah and Joseph parameters the tail exponent $\mu$ and one derived from the temporal behaviour such as power spectral $\beta$ at the other's expense. This poster will first describe recent work 1 in which we applied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both effects-to give insight into space physics data. I will show how we have resolved some contradictions seen in earlier work, where purely Joseph or Noah descriptions had been sought. Such hybrid Noah-Joseph ambivalent 2 behaviour is highly topical in physics. It is typically studied in the paradigm of the continuous time random walk CTRW rather than LFSM. Intriguingly the self-similarity exponent extracted from the CTRW differs from that seen in LFSM, being a ratio of $\mu$ and a temporal exponent rather than an additive function. The poster will elucidate the physical differences between these two pictures with reference to a newly-derived diffusion equation for LFSM, which replaces the second order spatial derivative in the equation of fBm 3 with a fractional derivative of order $\mu$. I will also show work in progress using an LFSM generator and simple analytic scaling arguments to study the problem of the area between a fractional L\'evy curve and a threshold-related both to Bernoulli excursions and to the burst size measure introduced by Takalo and Consolini into solar-terrestrial physics and further studied by Freeman et al 4,5. Finally I will discuss how LFSM gives the appearance of multi-affine scaling without having an underlying turbulent cascade or other multiplicative process. The importance of this property for the interpretation of natural time series will be discussed. 1 Watkins et al, Space Sci. Rev. 121, 271, 2005.\\ 2 Brockmann et al, Nature 439, 462, 2006.\\ 3 Wang and Lung, Phys. Lett. A 151, 119, 1990.\\ 4 Freeman et al, Geophys. Res. Lett. 27, 1367, 2000.\\ 5 Freeman et al, Phys. Rev. E 62, 8794, 2000.